Question 147744
solve the absolute calue inequality. other than 0, use interval notation to express the solution set and graph the solution set on a number line
|2y+6/3|<2
<pre><font size = 4 color = "indigo"><b>

Here are the rules for removing the absolute value bars in an
absolute value inequality of the form 

1.  For cases of the form {{{abs(expression)<A}}} where A is 
    a positive number, write this:

                      {{{-A<expression<A}}}     
                      
    
2.  For cases of the form {{{abs(expression)<=A}}} where A is a
    positive number, write this:

                      {{{-A<=expression<=A}}}     
                      

3.  For cases of the form {{{abs(expression)>A}}} where A is a 
    positive number, write this:
             
               {{{-A<expression}}}{{{OR}}}{{{expression>A}}}  

4.  For cases of the form {{{abs(expression)>=A}}} where A is a
    positive number, write this:
             
               {{{-A<=expression}}}{{{OR}}}{{{expression>=A}}} 

Your problem:

{{{abs((2y+6)/3)<2}}} is case 1, so to remove the absolute value symbols,
we write:

{{{-2<(2y+6)/3<2}}}

Now we multiply all three sides by 3 to clear of the fraction:

{{{-2*3<3*(2y+6)/3<2*3}}}

{{{-2*3<cross(3)*(2y+6)/cross(3)<2*3}}}

{{{-6<2y+6<6}}}

Subtract {{{6}}} from all three sides:

{{{-6-6<2y+6-6<6-6}}}

{{{-12<2y<0}}}

Divide all three sides by 2 to solve for y
in the middle:

{{{(-12)/2<(2y)/2<0/2}}}

{{{(-12)/2<(cross(2)y)/cross(2)<0/2}}}

{{{-6<y<0}}}

Mark -6 and 0 with open circles

<--------------o-----------------o------------>
 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  

Now shade between them:

<--------------o=================o------------>
 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1  0  1  2  3  4

In interval notation that is written (-6,0)

Edwin</pre>